In the above expression, \(J_2(\vert k_\perp \vert \vert r_\perp \vert )\)  is the Bessel function of the first kind. The following analysis is similar to that for gluon distributions in coordinate space. For \(k^2_\perp \gg Q^2_{sg}\) , the dominated contribution comes from the small dipole size \(\vert r_\perp \vert \ll 1/Q_{sg}\) , and can be evaluated by expanding the exponential term, then \(xG(x_g, k_\perp )\simeq \frac{C_F S_\perp }{4\alpha _s \pi ^3}\frac{Q^2_{sg}}{k^2_\perp }\)  and \(xH(x_g, k_\perp )\simeq \frac{C_F S_\perp }{4\alpha _s \pi ^3} \frac{Q^2_{sg}}{k^2_\perp }\) , in this limit, the two gluon distributions are completely same and behave as \(\frac{Q^2_{sg}}{k^2_\perp }\) . The perturbative property is agreement with other QCD results [1]}, [2]}, [3]}, [4]}. Moreover, due to \(xG(x_g, k_\perp )\simeq xH(x_g, k_\perp )\simeq \frac{C_F N_c A}{2\pi ^3 k^2_\perp }\) , the two gluon distributions scale like \(A\)  and have no nuclear shadowing effects, therefore, each of nucleon can additively give a contribution to the distributions in the dilute region. This conclusion is also agreement with that in coordinate space at small \(r_\perp \)  limit. In addition, \(\frac{xH(x_g, k_\perp )}{xG(x_g, k_\perp )}\approx 1\)  when \(k_\perp \)  is large enough, which gives a saturated boundary for the linearly polarized WW gluon distribution in the dilute regime [5]}. For \(\Lambda ^2_{\text{QCD}}\le k^2_\perp \ll Q^2_{sg}\) , the dominant contribution comes from large distances \(r_\perp \gg 1/Q_{sg}\) , one can neglect the exponential and the logarithmic \(r_\perp \)  terms, and arrive at \(xG(x_g, k_\perp )\simeq \frac{C_F S_\perp }{2\pi ^3\alpha _s}\ln \frac{Q^2_{sg}}{k^2_\perp }\)  and \(xH(x_g, k_\perp )\simeq \frac{N_c C_F A}{2\pi ^3}\frac{1}{Q^2_{sg}}\ln \frac{Q^2_{sg}}{k^2_\perp }\) , one can check that the two gluon distributions scale like \(A^{2/3}\) , which means the two gluon distributions have strong nuclear shadowing effects in the dense regime. Another interesting feature is the linearly polarized WW gluon distribution is weakly dependent on \(k_\perp \)  and approximates to a constant in this region. Furthermore, one can find \(\frac{xH(x_g, k_\perp )}{xG(x_g, k_\perp )}\)  is suppressed in the small \(k_\perp \)  limit, in which the multi-gluon scattering effects play a role that constrains the gluons to be polarized.
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